Question

question 2 of 10
when the factors of a trinomial are (x + p) and (x - q) then the coefficient of the x-term in the trinomial is:
a. the product of p and -q
b. the difference of p and -q
c. the sum of p and -q
d. the quotient of p and -q

Explanation:

Step1: Expand the factors

We have the factors \((x + p)\) and \((x - q)\). To find the trinomial, we expand the product using the distributive property (FOIL method):

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Step2: Identify the x - term coefficient

In the expanded trinomial \(x^2 + (p - q)x - pq\), the coefficient of the \(x\) - term is \(p - q\), which is the difference of \(p\) and \(q\) (or we can think of it as the sum of \(p\) and \(-q\) since \(p - q=p+(-q)\)). Wait, let's re - examine the options. The options are about \(p\) and \(-q\). Let's re - do the expansion with \(p\) and \(-q\) in mind. Wait, the factors are \((x + p)\) and \((x - q)\), so when we expand \((x + p)(x - q)=x^{2}-qx+px - pq=x^{2}+(p - q)x - pq\). The coefficient of \(x\) is \(p - q\), which is equivalent to \(p+(-q)\), that is the sum of \(p\) and \(-q\) (because \(p - q=p+(-q)\)). Wait, no, \(p - q\) is also the difference between \(p\) and \(q\), but let's check the options again. The options are:
A. The product of \(p\) and \(-q\) (that's \(-pq\), which is the constant term)
B. The difference of \(p\) and \(-q\) (that's \(p-(-q)=p + q\), not the \(x\) - term coefficient)
C. The sum of \(p\) and \(-q\) (since \(p+(-q)=p - q\), which is the coefficient of the \(x\) - term)
D. The quotient of \(p\) and \(-q\) (that's \(\frac{p}{-q}=-\frac{p}{q}\), not related)

Wait, maybe I made a mistake in the first analysis. Let's re - expand \((x + p)(x - q)\):

Using the formula \((a + b)(c + d)=ac+ad+bc+bd\), here \(a = x\), \(b = p\), \(c = x\), \(d=-q\)

So \((x + p)(x - q)=x\times x+x\times(-q)+p\times x+p\times(-q)=x^{2}-qx + px-pq=x^{2}+(p - q)x - pq\)

The coefficient of \(x\) is \(p - q\), and \(p - q=p+(-q)\), so it's the sum of \(p\) and \(-q\) (because \(p+(-q)=p - q\)).

Answer:

C. The sum of \(p\) and \(-q\)